Question 1

What are the steps required to estimate an ARCH/GARCH model?
A) First specify the appropriate equations for the correlation and the variance, then specify LLF and the computer will generate parameter values that maximise the LLF
B) First specify the appropriate equations for the median and the variance, then specify LLF and the computer will generate parameter values that maximise the LLF
C) First specify the appropriate equations for the mean and the variance, then specify LLF and the computer will generate parameter values that maximise the LLF
D) None of the above

Accepted Answer

In estimating ARCH/GARCH models, the first step involves specifying the equations for the mean and the variance of the time series. Then, the log-likelihood function (LLF) is specified, and the computer uses this to generate parameter values that maximize the LLF, fitting the model to the data.

Question 2

Suppose that a researcher wanted to obtain an estimate of realised ("actual") volatility. Which one of the following is likely to be the most accurate measure of volatility of stock returns for a particular day?
A) The price range (high minus low) on that day
B) The squared return on that day
C) The sum of the squares of hourly returns on that day
D) The squared return on the previous day

Accepted Answer

The sum of the squares of hourly returns on a particular day provides a more granular and thus potentially more accurate measure of the day's volatility, capturing intraday price movements that other methods might miss.

Question 3

-Which of the following statements are true regarding volatility:
(I) It measures the total risk of financial assets
(II) It can be used in computing value-at-risk
(III) It is a component of the Black-Scholes formula for deriving the prices of traded options
(IV) It can be estimated using the variance of asset returns
A) I only
B) I and II only
C) I, II and III only
D) I, II, III and IV

Accepted Answer

Volatility measures the degree of variation of a trading price series over time, which is a key component in various financial models and risk management techniques. It is used in computing value-at-risk (VaR), a risk management tool. It is also a crucial input in the Black-Scholes formula, which is used for pricing options. Volatility can indeed be estimated using the variance of asset returns, which quantifies the dispersion of returns around the mean.

Question 4

Suppose that a researcher estimates a GARCH(1,1) model and obtains a log likelihood function (LLF) value of 71.22. He is interested in testing whether an ARCH(1) model is a better model at describing volatility. If he estimates a model which imposes the necessary restrictions and obtains an LLF value of 68.21, what would be the conclusion of his likelihood ratio test (assuming a 5% significance level)?
A) Statistical evidence suggesting that ARCH(1) is better than GARCH(1,1)
B) Statistical evidence suggesting that ARCH(1) is not better than GARCH(1,1)
C) Statistical evidence suggesting that GARCH(1,1) is better than ARCH(1)
D) We cannot say because we would need to know the number of observations

Accepted Answer

The likelihood ratio test is calculated as 2*(LLF of unrestricted model - LLF of restricted model) = 2*(71.22 - 68.21) = 6.02. This value would then be compared to a chi-square distribution with degrees of freedom equal to the difference in the number of parameters between the models. Since GARCH(1,1) has more parameters than ARCH(1), the test assesses whether the simpler model (ARCH(1)) significantly reduces the fit of the model. A higher LLF for GARCH(1,1) suggests it is a better fit, and the calculated test statistic would need to be compared to a critical value from the chi-square distribution to make a final decision. However, without knowing the exact critical value (which depends on the degrees of freedom and is not provided), the general principle that a higher LLF indicates a better model leads to the conclusion that there is not statistical evidence suggesting ARCH(1) is better than GARCH(1,1).

Question 5

What would typically be the shape of the news impact curve for a series that exactly followed a GARCH(1,1) process?
A) It would be asymmetric, with a steeper curve on the left than the right
B) It would be asymmetric, with a steeper curve on the right than the left
C) It would be symmetric about zero
D) It would be discontinuous about zero

Accepted Answer

The news impact curve for a series that exactly follows a GARCH(1,1) process is symmetric about zero. This is because the GARCH model, in its standard form, treats positive and negative shocks of the same magnitude equally, leading to a symmetric volatility response.

Question 6

GJR and EGARCH are types of GARCH models that allow for:
A) An asymmetric response of returns to positive and negative shocks in the dependent variable
B) An asymmetric response of returns to positive and negative shocks to its lagged values
C) A symmetric response of volatility to positive and negative shocks
D) An asymmetric response of volatility to positive and negative shocks

Accepted Answer

GJR and EGARCH models are designed to capture the asymmetric response of volatility to positive and negative shocks, unlike the basic GARCH model which assumes a symmetric response. This means they can model the phenomenon where, for example, negative news impacts market volatility differently than positive news.

Question 7

Which of the following is the most plausible test regression for determining whether a series y contains "ARCH effects"?
A) 
B) 
C) 
D)

Accepted Answer

The answer of Which of the following is the most...

Question 8

What are the names of the following models?
(I)  (II)  (III)  (IV) 
A) GARCH (1), ARCH (1,1), GARCH(q) and ARCH(p,q), respectively
B) ARCH(1), GARCH(1,1), ARCH(q) and GARCH(p,q), respectively
C) ARCH(1), EGARCH(1,1), ARCH(q) and EGARCH(p,q), respectively
D) EGARCH (1), ARCH(1,1), EGARCH (q) and ARCH(p,q), respectively

Accepted Answer

The answer of What are the names of the following...

Question 9

Consider the three approaches to conducting hypothesis tests under the maximum likelihood framework. Which of the following statements are true?
(I) The Wald test is based on estimation only under the null hypothesis
(ii) The likelihood ratio test is based on estimation under both the null and the alternative hypotheses
(iii) The lagrange multiplier test is based on estimation under the alternative hypothesis only
(iv) The usual t and F-tests are examples of Wald tests
A) (ii) and (iv) only
B) (i) and (iii) only
C) (i), (ii), and (iv) only
D) (i), (ii), (iii), and (iv)

Accepted Answer

The likelihood ratio test involves estimation under both the null and alternative hypotheses, and the usual t and F-tests are examples of Wald tests.

Question 10

What is an appropriate approach to testing for 'ARCH effects'?
A) Run a regression, collect the residuals, regress the squared residuals on their lags and conduct a hypothesis test to check whether the coefficients of the lagged squared residuals are equal to zero
B) Run a regression, collect the fitted values, regress the fitted values on their squared lags and conduct a hypothesis test to check whether the coefficients of the lagged squared fitted values are equal to zero
C) Employ White's test
D) All of the above

Accepted Answer

Testing for ARCH (Autoregressive Conditional Heteroskedasticity) effects involves running a regression to obtain residuals, then regressing the squared residuals on their own lags, and finally conducting a hypothesis test to check if the coefficients of these lagged squared residuals are significantly different from zero. This process is aimed at identifying any patterns in the volatility (heteroskedasticity) that are dependent on past values.

Question 11

Which of these cannot be used to test for non-linearity?
A) Portmanteau tests
B) White test
C) Ramsey's RESET test
D) The BDS test

Accepted Answer

The answer of Which of these cannot be used to...

Question 12

Which of the following would represent the most appropriate definition for implied volatility?
A) It is the volatility of the underlying asset's returns implied from the price of a traded option and an option pricing model
B) It is the volatility of the underlying asset's returns implied from a statistical model such as GARCH
C) It is the volatility of an option price implied from a statistical model such as GARCH
D) It is the volatility of an option price implied from the underlying asset volatility

Accepted Answer

Implied volatility is the market's forecast of a likely movement in a security's price and is implied from the price of a traded option using an option pricing model, such as the Black-Scholes model. It is not directly derived from statistical models like GARCH (which is used for estimating the volatility of returns of the underlying asset, not the option) or from the volatility of the option price itself.

Question 13

Assume that you have estimated a GJR model of monthly stock returns and you obtain the following equations:   Suppose that , what would be the fitted conditional variance for time t if and then if ?   
A) 1.62 and 1.67, respectively
B) 1.64 and 1.59, respectively
C) 1.59 and 1.64, respectively
D) 1.67 and 1.62, respectively

Accepted Answer

The answer of Assume that you have estimated a GJR...

Question 14

Volatility clustering is
A) The tendency for financial asset returns to have distributions that exhibit fat tails
B) The tendency for financial asset return volatility to appear in bunches
C) The tendency for volatility to rise more following a large price fall than following a price rise of the same magnitude
D) All of the above

Accepted Answer

Volatility clustering refers to the phenomenon where large changes in asset prices are followed by large changes, and small changes are followed by small changes, indicating that volatility tends to cluster over time.

Question 15

Which of the following are NOT features of an IGARCH(1,1) model?
(I) Forecasts of the conditional variance will converge upon the unconditional variance as the horizon tends to infinity
(ii) The sum of the coefficients on the lagged squared error and the lagged conditional variance will be unity
(iii) Forecasts of the conditional variance will decline gradually towards zero as the horizon tends to infinity
(iv) Such models are never observed in reality
A) (ii) only
B) (ii) and (iv) only
C) (ii), (iii) and (iv) only
D) (i), (ii), (iii) and (iv)

Accepted Answer

(ii) is incorrect because the sum of the coefficients on the lagged squared error and the lagged conditional variance will be less than unity.

Question 16

Which one of the following problems in finance could not be usefully addressed by either a univariate or a multivariate GARCH model?
A) Producing option prices
B) Producing dynamic hedge ratios
C) Producing time-varying beta estimates for a stock
D) Producing forecasts of returns for use in trading models
(e) Producing correlation forecasts for value at risk models

Accepted Answer

GARCH models are primarily used for modeling volatility and not directly for forecasting returns. They are useful for options pricing, dynamic hedging, estimating time-varying betas, and forecasting correlations for risk models, but not for directly forecasting returns.

Question 17

Which of the following is true about ARCH and GARCH models?
(I) They are used for modelling and forecasting volatility
(II) They are non-linear models
(III) They can both be estimated using OLS
(IV) Series estimated using these models must have a unit root process
A) I only
B) I and II only
C) I, II and III only
D) I, II, III and IV

Accepted Answer

ARCH (Autoregressive Conditional Heteroskedasticity) and GARCH (Generalized Autoregressive Conditional Heteroskedasticity) models are indeed used for modeling and forecasting volatility, making them particularly useful in financial time series analysis where volatility clustering is common. They are non-linear in the sense that they model the variance of the current error term as a function of the past squared errors (ARCH) or both past squared errors and past variances (GARCH), which introduces non-linearity. However, they cannot be estimated using Ordinary Least Squares (OLS) due to this non-linearity; instead, methods like Maximum Likelihood Estimation (MLE) are used. Additionally, the requirement for a unit root process is not a prerequisite for ARCH or GARCH models; they are designed to model volatility regardless of the presence of a unit root in the series.

Question 18

Which of these is an appropriate technique used in estimating models from the GARCH family?
A) Maximum likelihood
B) Instrumental variables
C) Indirect least squares
D) Ordinary least squares

Accepted Answer

Maximum likelihood estimation is the appropriate technique for estimating models from the GARCH (Generalized Autoregressive Conditional Heteroskedasticity) family. This method is used because it can handle the time-varying volatility that is characteristic of GARCH models.

Question 19

Consider the following conditional variance equation for a GJR model. h_t = $\alpha$₀ + $\alpha$₁+$\beta$h_t_-1+$\gamma$u_t_-1²I_t_-1 where I_t_-1 = 1 if u_t_-1 < 0 = 0 otherwise For there to be evidence of a leverage effect, which one of the following conditions must hold? A) $\alpha$₀ positive and statistically significant B) $\gamma$ positive and statistically significant C) $\gamma$ statistically significantly greater than $\alpha$₀ D) $\alpha$₁+$\beta$ statistically significantly less than $\gamma$

Accepted Answer

The answer of Consider the following conditional variance equation for...

Quiz 10: Switching Models

What are the steps required to estimate an ARCH/GARCH model?

Suppose that a researcher wanted to obtain an estimate of realised ("actual") volatility. Which one of the following is likely to be the most accurate measure of volatility of stock returns for a particular day?

What would typically be the shape of the news impact curve for a series that exactly followed a GARCH(1,1) process?

GJR and EGARCH are types of GARCH models that allow for:

Which of the following is the most plausible test regression for determining whether a series y contains "ARCH effects"?

What are the names of the following models? (I) (II) (III) (IV)

What is an appropriate approach to testing for 'ARCH effects'?

Which of these cannot be used to test for non-linearity?

Which of the following would represent the most appropriate definition for implied volatility?

Assume that you have estimated a GJR model of monthly stock returns and you obtain the following equations: Suppose that , what would be the fitted conditional variance for time t if and then if ?

Volatility clustering is

Which one of the following problems in finance could not be usefully addressed by either a univariate or a multivariate GARCH model?

Which of the following is true about ARCH and GARCH models? (I) They are used for modelling and forecasting volatility (II) They are non-linear models (III) They can both be estimated using OLS (IV) Series estimated using these models must have a unit root process

Which of these is an appropriate technique used in estimating models from the GARCH family?

Consider the following conditional variance equation for a GJR model. ht = α\alphaα 0 + α\alphaα 1+ β\betaβ ht-1+ γ\gammaγ ut-12It-1 where It-1 = 1 if ut-1 < 0 = 0 otherwise For there to be evidence of a leverage effect, which one of the following conditions must hold?